Lower Closure is Prime Ideal for Prime Element

Theorem

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a lattice.

Let $p \in S$ be a prime element.

Then $p^\preceq$ is a prime ideal.

Proof

Let $x, y \in S$ such that

$x \wedge y \in p^\preceq$

By definition of lower closure of element:

$x \wedge y \preceq p$

By Characterization of Prime Ideal:

$x \preceq p$ or $y \preceq p$

Thus by definition of lower closure of element:

$x \in p^\preceq$ or $y \in p^\preceq$

$\blacksquare$

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_7:33